Subgradient Descent Learns Orthogonal Dictionaries
This addresses a fundamental representation learning problem for machine learning, offering a simpler and more efficient provable method.
The paper tackles dictionary learning by proving that subgradient descent with random initialization can recover orthogonal dictionaries under mild statistical assumptions, contrasting with previous methods that required expensive computation or delicate initialization.
This paper concerns dictionary learning, i.e., sparse coding, a fundamental representation learning problem. We show that a subgradient descent algorithm, with random initialization, can provably recover orthogonal dictionaries on a natural nonsmooth, nonconvex $\ell_1$ minimization formulation of the problem, under mild statistical assumptions on the data. This is in contrast to previous provable methods that require either expensive computation or delicate initialization schemes. Our analysis develops several tools for characterizing landscapes of nonsmooth functions, which might be of independent interest for provable training of deep networks with nonsmooth activations (e.g., ReLU), among numerous other applications. Preliminary experiments corroborate our analysis and show that our algorithm works well empirically in recovering orthogonal dictionaries.